Mathematical Sciences: Presidential Young Investigator Award

这位总统青年研究员的数学研究的主要重点将是几何问题引起的偏微分方程式。该方程是典型的椭圆型和抛物型非线性方程。其中一类问题涉及到所谓的曲线缩短。问题是描述一族平面曲线的长时间行为，这些曲线不断演变，使它们的法向速度与它们的曲率重合。当法向速度一般取决于曲率和时间时，会出现更复杂的情况；后一种情况发生在二维相变模型中。曲线缩短问题的解已被证明在某些情况下会产生奇点(爆破)。(适当调整)奇异部分的渐近形状不依赖于初始数据。数值模拟表明，曲率的实际爆破率是时间的重对数的函数。将开展工作，以确定这一观察结果是否属实，并寻求对估计数中的对数项负责的机制。在确定哈密顿系统在光滑紧致曲面的余切丛上的指定能量的周期轨道方面也将做工作。假设哈密顿量是凸的，并且在每根纤维上都是均匀的。这项工作的一个具体目标是确定能量面上存在周期轨道，其在流形上的投影与给定周期轨道的投影相交规定次数。这样的结果可以看作是Poincare-Birkhoff定理的整体类比。相关工作将研究两维哈密顿系统的部分解之间的联系。估计连接数可以被视为类似于测量标量抛物型偏微分方程根的数目。

The primary focus of the mathematical research of this Presidential Young Investigator will be on partial differential equations arising from geometrical problems. The equations are typically nonlinear of elliptic and parabolic type. One class of problems involves what is known as curve shortening. The question is one of describing the long time behavior of a family of plane curves which evolve so that their normal velocity coincides with their curvature. More complicated situations arise when the normal velocity depends in a general way on the curvature and time; the latter case occurring in models of phase transitions in two dimensions. The solution to a curve shortening problem has been shown to develop singularities (blow-up) under certain circumstances. The asymptotic shape of the (suitably rescaled) singular part does not depend on the initial data. Numerical simulations suggest that the actual blow-up rate of the curvature is a function of the iterated logarithm of time. Work will be done in determining if this observation is true and in seeking the mechanism responsible for the log-term in the estimate. Work will also be done in determining periodic orbits with prescribed energy of Hamiltonian systems on cotangent bundles of smooth compact surfaces. The Hamiltonian is assumed to be convex and even on each fibre. A particular goal of the work is to establish the existence of periodic orbits on energy surfaces whose projections on the manifold intersect the projection of a given periodic orbit a prescribed number of times. Such a result may be regarded as a global analog of the Poincare-Birkhoff theorem. Related work will study the linking of parts of solutions of Hamiltonian systems in two dimensions. Estimating linking numbers can be viewed as analogous to measuring the number of roots of a scalar parabolic partial differential equation.